The fierce fighting in Beirut spawned efforts in a number of capitals last week to find a solution to the crisis. These bids came before reports by Lebanese government sources last night that Palestine Liberation Organization Chairman Yasser Arafat had agreed in principle to lay down the guerrillas' arms and leave Lebanon with his troops. The earlier proposals by the five major actors emerged in public statements and private communications. They were:

* ISRAEL--The Cabinet demanded yesterday that all Palestinian guerrillas in the city surrender their arms to the Lebanese Army and leave for Syria. The demand seemed to represent a switch from Israel's position earlier in the week, when high officials said that only the leadership and "core" of the PLO had to leave Lebanon but that they had to go to a country that does not border on Israel.

The Cabinet did not fix a deadline for the departure of the estimated 5,000 to 6,000 guerrillas but pledged to observe Friday evening's cease-fire for an indefinite period, apparently to allow time for U.S. special envoy Philip C. Habib to try to convince the PLO to give up peacefully and leave. If Habib failed, however, Israeli sources said the government planned to order an attack on the PLO stronghold and seek to crush the organization militarily.

* PLO--Salah Khalaf, a senior lieutenant of Palestine Liberation Organization leader Yasser Arafat, unveiled on Monday the group's proposal for its troops to return to their refugee camp bases on the outskirts of Beirut in exchange for a six-mile Israeli withdrawal from the capital and a reopening of the Beirut-to-Damascus road. The Lebanese Army would be deployed around the PLO camps as a buffer force, and negotiations would open some time later between the guerrillas and the Lebanese government to determine the PLO's future status in the country.

* UNITED STATES--There were at least two tracks for U.S. proposals during the week, and the differences between them caused confusion and in the end may have encouraged the PLO to think that it could escape without accepting Israel's terms, Washington Post correspondent Jonathan C. Randal reported.

1. The Habib Track. Habib met Lebanon's National Salvation Council, which groups the nation's major political and military factions, on Wednesday. Apparently acting on orders from Secretary of State Alexander M. Haig Jr., Habib implicitly rejected the PLO's proposals in such a way as to leave the Lebanese with serious questions whether he was speaking for the United States or for Israel. Habib said he wanted "very specific answers" to these questions: did the PLO intend to maintain a military presence in Lebanon; did its leaders intend to remain in Beirut, and would the Palestinians in Lebanon (as distinct from the PLO) accept the sovereignty of the Lebanese state? He clearly was looking for "no" answers to the first two questions and a "yes" for the third, and leftist Druze leader Walid Jumblatt said the United States was demanding "total surrender" by the PLO.

2. The Clark Track. Later Wednesday, however, Arafat received a phone call from Saudi King Fahd assuring him that the United States had promised an Israeli withdrawal of three miles, Palestinian sources said. The king said the undertaking had been made by National Security Adviser William P. Clark, and the PLO apparently believed the Saudis were working successfully on its behalf. A PLO spokesman went out of his way the next day to spare Habib any criticism and suggested that the envoy was 24 hours behind developments in Washington, apparently referring to the Clark message.

The Haig-Habib track seemed to be in ascendancy on Friday when PLO representatives in France and India forwarded to Arafat messages that their host governments had received from the U.S. State Department reiterating Haig's original tough terms. Later that day, Habib told Lebanese Prime Minister Shafiq Wazzan that the Palestinians should capitulate, lay down their arms and quit Lebanon unconditionally.

* LEBANON--Longstanding divisions among Lebanon's different political and religious factions stymied consensus on a single position. There was vague talk of deploying the Lebanese Army as a buffer between the Israelis and Palestinians, but so far it has come to naught.

The leader of the pro-Israeli Christian Phalangist militia, Bashir Gemayel, offered an olive branch to Arafat Tuesday by suggesting that he be invited to a council meeting, but Gemayel later backed off. For its part, the Moslem establishment refused to join the Phalangists in repudiating the Palestinians when Moslem prime minister Wazzan rejected what he called Habib's "military blackmail" in calling for a PLO surrender.

* FRANCE--After consultations with the PLO in Beirut, Paris proposed a resolution in the U.N. Security Council Friday offering terms that were similar to those that the PLO had suggested earlier in the week. The resolution called for a withdrawal of Israeli forces to six miles from Beirut and the simultaneous withdrawal of Palestinian guerrillas to their bases on the edge of the city. France deliberately left vague whether the PLO would surrender its arms.